Fundamentals of Astrodynamics and Applications

1135 Pages·2013·14.911 MB·English

by  Vallado, David A.

Common Two-body Equations
Specific Angular Momentum h = r×v
.
h = mp = rvCOS(ffpa) = r2n = rava = rpvp
Specific Mechanical Energy y = -v---2-–m---- = –--m----
2 r 2a
Orbital Period = 2p a----3-
m
m
v2–----r–(r⋅v)v
Eccentricity Vector e = ----------------r--------------------------------
m
p
Radius r = 1-----+-----e----C----O---S---(---n----)-
.
Radial Rate r. = --r---n----e-----S---I-N----(---n----)---
1+eCOS(n)
n· = n----a---2-- 1–e2
r2
Semi-parameter p = h----2-
m
Semi-major axis a = m2----p----21/3 = 3 n-m---2-
continued on next page
Circle Ellipse Parabola Hyperbola
14.0
3.0 5.0 10.0
v
v
r b p r v p p v
r r
0.0 F, F′ 0.0 F′ ra F rp 0.0 F′ F 0.0 F rp F′
c rp −a
a
a, r, p –5.0
–3.0–3.0 0.0 3.0 ER –9.0 0.0 4.0 ER–10.–012.0 0.0 8.0 –14.0
Eccentric Anomaly Undefined En+1 = En+M-------–1----E-–---n-e----+C---O-e--S--S--(-I-EN----n(---E)----n---)- Hn+1 = H–1n0.0+M-------e–---C-e---O-S--S-I0-N-.H0---H--(--(-H-H---n--n-)--)--–-+---1-1-0H-.-0---n--
CSOINS((EE)) == --1S--e--1--I--+N--+----+--(--e----nCe--C----O)--C--O--S--O--S----(1--S(--n----n(--–--)n----)--e-)--2-- B = TANn-2-- CSOINSHH((HH)) == --1S--e--1--I--+N--+----+--(--e----nCe--C----O)--C--O--S--O--S----(e--S(--n--2--n(----)n--–--)---)--1--
True Anomaly Undefined SIN(n) = -S--1-I-N---–-(---Ee----)C---O---1-S---(-–-E---e-)--2-- SIN(n) = p---r-B--- SIN(n) = –----S-1--I-N-–---H--e--(-C-H---O--)-S---H---e-(--2H----–-)---1--
COS(n) = -1--C--–-O---S-e---(C--E-O---)-S---–(--E--e----) COS(n) = p-----–r----r-- COS(n) = 1---C--–-O---Se---H-C---(O---H-S--H-)----(–--H--e---)-
Flight Path Angle ffpa = 0° SIN(ffpa) = -----1----–----e-e--S-2--I-CN---O-(---SE---2-)---(--E-----)--- fparabolic = n-2-- SIN(ffpa) = -----e---2-–--C-e--O--S-S-I--NH---H-2---((--H-H----)-)---–----1--
COS(ffpa) = 1-----–-----e-1-2----C–---O-e--S-2--2---(---E----)- COS(ffpa) = e---2----C---O----Se--H-2---2--–--(--1-H----)-----–----1--
Circle Ellipse Parabola Hyperbola
Equation r = a r = -1----+-----e-----Cp---O---S---(---n----)- r = p2---(1+B2) r = -1----+-----e-----Cp---O---S---(---n----)-
Perigee rp = a rp = a(1 − e) rp = p/2 rp = a(1 − e)
Apogee ra = a ra = a(1 + e) ra = a ra = a(1 + e)
Semi-parameter p = a p = a(1 − e2) p = –-hm---2- p = a(1 − e2)
m
Semi-major axis a = r a = –2----y--- a = ∞ a = –2--m--y---
Velocity v = m-r--- v = 2m-r---+y = -2--r-m---–-ma--- = m-r---2–1-----+----1-e---–C----O-e--S-2--(---n----)- vparabola = vesc = -2--r-m--- v = -2--r-m---–-ma---
2yh2
Eccentricity e = 1+-------------
m2
Mean Motion n = 2----p---- = a-m---3- n = 2----p---- = a-m---3- np = 2 p-m---3- n = 2----p---- = a-m---3-
Mean anomaly Undefined M = E − eSIN(E) -B-3--3--+B–np(Δt) = 0 M = eSINH(H) − H
Period = 2p a-m---3- = 2p a----3-
m
Velocity and Rate Formulations
Tangential Range
Radial Range v2
Tangential Velocity Dv
Dr
Radial Velocity
r1 r2 Dv v1
Dr
Range Rates Velocity Rates
Range Dr = r2–r1 Velocity Dv = v2–v1
Radial Range rr = r---1----⋅---D----r- Radial Velocity rv = -D----r----⋅---D----v--
r
r 1
1
Tangential Range tr = (Dr)2–rr2 Tangential Velocity tv = (Dv)2–rv2
Dr×Dv
Angular Rate v = --------------------
vel Dr⋅Dr
v = TAN –1----------t--v------------
 
vel Dr +rvt
Acceleration tmust first be specified
Dr×Da 2(Dr⋅Dv)(Dr×Dv)
v = --------------------–---------------------------------------------------- Angular acceleration
accel Dr⋅Dr (Dr)4
FUNDAMENTALS OF ASTRODYNAMICS AND APPLICATIONS
Fourth Edition
THE SPACE TECHNOLOGY LIBRARY
Published jointly by Microcosm Press and Springer
An Introduction to Mission Design for Geostationary Satellites, J. J. Pocha
Space Mission Analysis and Design, 3rd edition, James R. Wertz and Wiley J. Larson
Space Mission Engineering, James R. Wertz
Handbook of Geostationary Orbits, E. M. Soop
Spacecraft Structures and Mechanisms, From Concept to Launch, Thomas P. Sarafin
Spaceflight Life Support and Biospherics, Peter Eckart
Reducing Space Mission Cost, James R. Wertz and Wiley J. Larson
The Logic of Microspace, Rick Fleeter
Space Marketing: A European Perspective, Walter A. R. Peeters
Orbit and Constellation Design and Management, James R. Wertz
Influence of Psychological Factors on Product Development, Eginaldo Shizuo Kamata
Essential Spaceflight Dynamics and Magnetospherics, Boris Rauschenbakh,
Michael Ovchinnikov, and Susan McKenna-Lawlor
Space Psychology and Psychiatry, Nick Kanas and Dietrich Manzey
Fundamentals of Space Medicine, Gilles Clément
Fundamentals of Space Biology, Gilles Clément and Klaus Slenzka
Microgravity Two-Phase Flow and Heat Transfer, Kamiel Gabriel
Artificial Gravity, Gilles Clément and Angelia Buckley
The Space Technology Library Editorial Board
Managing Editor: James R. Wertz, Microcosm, Inc., Hawthorne, CA
Editorial Board: Roland Doré, International Space University, Strasbourg, France
Wiley J. Larson, United States Air Force Academy
Tom Logsdon, Rockwell International (retired)
Landis Markley, Goddard Space Flight Center
Robert G. Melton, Pennsylvania State University
Keiken Ninomiya, Institute of Space & Astronautical Science, Japan
Jehangir J. Pocha, Matra Marconi Space, Stevenage, England
Gael Squibb, Jet Propulsion Laboratory
Martin Sweeting, University of Surrey, England
About the Author
David Vallado is currently a Senior Research Astrodynamicist with Analytical Graphics, Inc. in
Colorado Springs, Colorado. He holds a BS degree in Astrodynamics Engineering from the U. S.
Air Force Academy and MS from the Air Force Institute of Technology. He also holds an MS in
Systems Management from the University of Southern California. He is a retired Air Force Lieu-
tenant Colonel. His Air Force experience includes: Stage 1 Project Manager for MX missile, SAC
Headquarters Analysis Division, U. S. Air Force instructor, Air Force Research Laboratory Scien-
tist, and Headquarters U. S. Space Command Analysis Office. He has also worked as a principal
engineer with Raytheon in Denver, Colorado.
Fundamentals of Astrodynamics
and Applications
Fourth Edition
David A. Vallado
Analytical Graphics Inc.
with technical contributions by
Wayne D. McClain
Charles Stark Draper Laboratory
Space Technology Library
Published by
Microcosm Press
Hawthorne, CA
Library of Congress Cataloging-in-Publication Data
A C.I.P. Catalogue record for this book is available from
the Library of Congress
ISBN 978-1881883180 (pb) (acid-free paper)
ISBN 978-1881883197 (hb) (acid-free paper)
ISBN 978-1881883203 (ebook) (acid-free paper)
Published by
Microcosm Press
4940 West 147th Street, Hawthorne, CA 90250-6708 USA
Sold and distributed
by Microcosm, Inc.
4940 West 147th Street, Hawthorne, CA 90250-6708 USA.
Printed on acid-free paper
All Rights Reserved
© 2013 David A. Vallado
First Printing, 2013
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording, or by any information storage and
retrieval system, without written permission from the publishers.
Printed in the United States of America
To God from whom all blessings and wisdom flow,
For the free gift of salvation through His true and only Son, Jesus Christ,
To the dedication of one’s life to the pursuit of wisdom,
To my parents for their devotion and example,
To my wife’s support through this long task,
and especially to my lovely children,
Simone, Kathleen, and Samuel